/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 25 The Taylors have purchased a $$\... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 25

The Taylors have purchased a $$\$ 270,000$$ house. They made an initial down payment of $$\$ 30,000$$ and secured a mortgage with interest charged at the rate of \(8 \%\) /year on the unpaid balance. Interest computations are made at the end of each month. If the loan is to be amortized over \(30 \mathrm{yr}\), what monthly payment will the Taylors be required to make? What is their equity (disregarding appreciation) after 5 yr? After 10 yr? After 20 yr?

Short Answer

Expert verified
The Taylors will have a monthly payment of approximately $$\$1,751.16$$ on their mortgage. Their equity (disregarding appreciation) after 5 years is approximately $$\$14,381.37$$, after 10 years is approximately $$\$35,595.98$$, and after 20 years is approximately $$\$99,478.05$$.

Step by step solution

01

Determine Loan Amount and Interest rate per month

First, subtract the initial down payment from the house price to find the mortgage loan amount: \(\$ 270,000 - \$ 30,000 = \$ 240,000\). The annual interest rate is \(8\%\), so to find the monthly interest rate, divide by 12: \(\frac{8}{100 \times 12} = 0.0066667\).
02

Calculate Monthly Payment

Use the following mortgage loan formula to calculate the monthly payment: \(M = P\frac{r(1+r)^n}{(1+r)^n-1}\), where \(M\) is the monthly payment, \(P\) is the loan amount (\$240,000), \(r\) is the monthly interest rate (0.0066667), and \(n\) is the total number of payments (30 years * 12 months/year = 360 payments). Plugging in the values, we get: \[M = 240000\frac{0.0066667(1+0.0066667)^{360}}{(1+0.0066667)^{360}-1} \approx \$1,751.16\]
03

Calculate Equity after 5, 10, and 20 years

To calculate the equity at specific years, first, we need to find the remaining balance on the mortgage loan after those years. To find the remaining balance on the mortgage loan, we can use the following formula: \(B = P\frac{(1+r)^n-(1+r)^k}{(1+r)^n-1}\), where \(B\) is the remaining balance, \(P\) is the loan amount (\$240,000), \(r\) is the monthly interest rate (0.0066667), \(n\) is the total number of payments (360), and \(k\) is the number of payments made so far. Equity after 5 years (\(k = 5 \times 12 = 60\)): \[B = 240000\frac{(1+0.0066667)^{360}-(1+0.0066667)^{60}}{(1+0.0066667)^{360}-1} \approx \$225,618.63\] Equity after 5 years: \(\$ 270,000 - \$ 30,000 - \$ 225,618.63 \approx \$ 14,381.37\] Equity after 10 years (\(k = 10 \times 12 = 120\)): \[B = 240000\frac{(1+0.0066667)^{360}-(1+0.0066667)^{120}}{(1+0.0066667)^{360}-1} \approx \$204,404.02\] Equity after 10 years: \(\$ 270,000 - \$ 30,000 - \$ 204,404.02 \approx \$ 35,595.98\] Equity after 20 years (\(k = 20 \times 12 = 240\)): \[B = 240000\frac{(1+0.0066667)^{360}-(1+0.0066667)^{240}}{(1+0.0066667)^{360}-1} \approx \$140,521.95\] Equity after 20 years: \(\$ 270,000 - \$ 30,000 - \$ 140,521.95 \approx \$ 99,478.05\] So, the Taylors' equity after 5 years is approximately \$14,381.37, after 10 years is approximately \$35,595.98, and after 20 years is approximately \$99,478.05.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Loan Amortization
Loan amortization is the process of gradually reducing a debt through periodic payments. In the context of a mortgage, this means paying off a loan over a set period, called the term of the loan. When you make a monthly mortgage payment, it covers two things: the interest due for the month and a portion that goes towards reducing the principal, the original amount borrowed.

Amortization schedules are comprehensive tables that detail each payment over the loan's life span, showing exactly how much of each payment reduces the principal and how much goes toward interest. At the beginning of the loan term, a higher portion of your payments goes toward interest with a smaller part reducing the principal. As time goes by, this gradually shifts, and later payments pay down more of the principal and less interest, reflecting the declining balance owed.
Monthly Mortgage Payments
Monthly mortgage payments are the regular amounts a borrower agrees to pay a lender in order to repay a mortgage loan. A portion of each payment goes towards the loan's interest, while the remainder pays down the principal balance. The size of the payments depends on the loan amount, the interest rate, and the term of the loan.

The formula for calculating the monthly payment is \[M = P\frac{r(1+r)^n}{(1+r)^n-1}\], where \(M\) is the monthly payment, \(P\) is the principal, \(r\) is the monthly interest rate, and \(n\) is the total number of payments. When you know these values, you can determine exactly what you'll need to pay each month over the course of the loan. In the case of the Taylors, their monthly payment was calculated to be approximately \$1,751.16.
Calculating Equity
Equity refers to the portion of your property's value that you actually own, as opposed to the portion still owed on a mortgage. It's calculated by taking the market value of the property and subtracting the remaining balance of the mortgage. Building equity is one of the principal benefits of homeownership because it represents the part of your home that you have 'paid off' and belongs to you, outright.

The Taylors' equity after varying lengths of time was calculated using the remaining balance formula in the original solution, showing how equity increases as the principal is paid down. Effectively, as they continue making payments, their equity grows because the owed principal (and consequently the interest charge) decreases 鈥� an advantageous financial position.
Interest Rate Calculations
Interest rates are a critical component of mortgage loans, representing the cost of borrowing. The interest rate is usually expressed as an annual percentage, but since mortgage payments are typically made on a monthly basis, the yearly rate is converted to a monthly rate by dividing by 12.

In the case of the Taylors, the annual interest rate of 8% became a monthly rate by dividing by 12, resulting in 0.0066667. When you multiply this monthly rate by the balance remaining on the loan, you get the amount of interest due for that month. As the loan progresses and the principal is paid down, the amount of interest charged each month decreases, which is well illustrated by the decreasing balance in the Taylors' loan over the years.

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